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Properties

Label 9.28.a
Level $9$
Weight $28$
Character orbit 9.a
Rep. character $\chi_{9}(1,\cdot)$
Character field $\Q$
Dimension $11$
Newform subspaces $5$
Sturm bound $28$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 28 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(28\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{28}(\Gamma_0(9))\).

Total New Old
Modular forms 29 12 17
Cusp forms 25 11 14
Eisenstein series 4 1 3

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)Dim
\(+\)\(5\)
\(-\)\(6\)

Trace form

\( 11 q - 16470 q^{2} + 556696244 q^{4} + 1234423260 q^{5} + 251450690980 q^{7} - 1666710502200 q^{8} + O(q^{10}) \) \( 11 q - 16470 q^{2} + 556696244 q^{4} + 1234423260 q^{5} + 251450690980 q^{7} - 1666710502200 q^{8} - 16087382296380 q^{10} - 254784665487240 q^{11} + 705831786552550 q^{13} - 5801664363745824 q^{14} + 7157300170423952 q^{16} + 68857660855028340 q^{17} + 57061287146645392 q^{19} + 266655528214427880 q^{20} + 1959553730964365640 q^{22} - 2533961125232515440 q^{23} + 8109788261025349145 q^{25} + 17058603439790252028 q^{26} + 34220099410483659520 q^{28} + 52453945344150636876 q^{29} + 116042588938258087828 q^{31} - 561175409135838808800 q^{32} + 47226968373154305708 q^{34} - 1315423534603461474240 q^{35} + 196851461564405236090 q^{37} - 4201114806339480199800 q^{38} + 12038299385786809834320 q^{40} - 15638521092134333700348 q^{41} + 36889189721563122268960 q^{43} - 104756047106148915906096 q^{44} + 144056995626044523134448 q^{46} - 136862891990032480770240 q^{47} + 303182924470378911177171 q^{49} - 422250090661382576117850 q^{50} + 409536396454455000853240 q^{52} - 883010940290266308321060 q^{53} + 927384568921053779154480 q^{55} - 2685824046230363941150080 q^{56} + 1063627588340557601934420 q^{58} - 804338659486113678147528 q^{59} - 8875048512622192162286 q^{61} + 2551663717072706323547280 q^{62} - 595625684392085963134912 q^{64} + 5328019715754387832263720 q^{65} - 7109562655363573314419000 q^{67} + 22014481906186865649110520 q^{68} - 25991449536603272240957760 q^{70} + 22342361331886652239431408 q^{71} - 51359204161990840134954470 q^{73} + 99467554834645954003882476 q^{74} - 53054891982212160632257328 q^{76} + 61065486271480588806706560 q^{77} - 84983108724974572376575820 q^{79} + 17526481096137663428883360 q^{80} - 42429082253923659849839460 q^{82} - 120310930120565401090791480 q^{83} + 356012222648950292110219080 q^{85} - 349240544924153279486053320 q^{86} + 807645665422969657454052000 q^{88} - 417422468435037785797887612 q^{89} - 135539861194419095540210392 q^{91} - 1133967156230260469638841760 q^{92} + 2089915194539924338736560320 q^{94} - 1380490101860775265387947600 q^{95} + 1075099931121814250588822290 q^{97} - 2520067233865801508632593990 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{28}^{\mathrm{new}}(\Gamma_0(9))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3
9.28.a.a 9.a 1.a $1$ $41.567$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-467920383820\) $+$ $N(\mathrm{U}(1))$ \(q-2^{27}q^{4}-467920383820q^{7}-2107227619093030q^{13}+\cdots\)
9.28.a.b 9.a 1.a $2$ $41.567$ \(\Q(\sqrt{30001}) \) None \(-21582\) \(0\) \(1771946100\) \(369665199904\) $-$ $\mathrm{SU}(2)$ \(q+(-10791-\beta )q^{2}+(101301922+\cdots)q^{4}+\cdots\)
9.28.a.c 9.a 1.a $2$ $41.567$ \(\Q(\sqrt{6469}) \) None \(-3168\) \(0\) \(4906065060\) \(-151657089584\) $-$ $\mathrm{SU}(2)$ \(q+(-1584-\beta )q^{2}+(2432512+3168\beta )q^{4}+\cdots\)
9.28.a.d 9.a 1.a $2$ $41.567$ \(\Q(\sqrt{18209}) \) None \(8280\) \(0\) \(-5443587900\) \(-175391963600\) $-$ $\mathrm{SU}(2)$ \(q+(4140-\beta )q^{2}+(95311648-8280\beta )q^{4}+\cdots\)
9.28.a.e 9.a 1.a $4$ $41.567$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(0\) \(0\) \(676754928080\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(73205452+11\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{28}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{28}^{\mathrm{old}}(\Gamma_0(9)) \cong \) \(S_{28}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{28}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)